Spectral-based numerical solution method of the incompressible Navier-Stokes equation in a stretched coordinate system

The numerical solution of the Navier-Stokes equation is widely used in computational fluid dynamics to address engineering problems related to fluids. The method can deal with nonlinear coupling between velocity and pressure fields. In this study, a robust numerical approach integrates the pseudospectral method with a splitting algorithm and a pressure correction technique to solve the two-dimensional incompressible Navier-Stokes equation effectively. This study aims to confirm the accuracy of the suggested method by comparing its results with benchmark two-dimensional solutions and to assess the computational efficiency of the developed algorithm. The pseudospectral method is typically applied to simpler problems involving Ordinary Differential Equations and Partial Differential Equations. Spatial discretization was performed using cardinal Chebyshev basic functions, which offer spectral accuracy and apply to non-periodic domains. The problem was divided into smaller, more manageable pseudospectral grids using a splitting algorithm and combined with a nonlinear term formulation. The combination technique allows for the decoupling of velocity and pressure computations. The numerical scheme was implemented in MATLAB and evaluated for Reynolds numbers (Re) of 100, 400, and 1000. The simulation results indicate a strong correlation between the predicted velocity profile and vortex formation, aligning closely with established benchmark data. The percentage differences of the compared data were 0.02% for U-Velocity and 2.11% for V-Velocity at Re = 100, respectively. The predicted flow pattern and centre location of the vortices matched closely with the reference value, confirming the method's accuracy. Additionally, the approach exhibited rapid convergence and computational efficiency. In conclusion, the proposed technique successfully demonstrates the capability to solve the incompressible flow problem and provides accurate results up to a Reynolds number of 1000. A stability analysis described in this study indicates that the proposed method remains applicable at higher Reynolds numbers.